RHCP and LHCP measurement

I have a dual band circularly polarized antenna operating at 1.8 GHz and 2.4 GHz. Can anybody tell as to how to practically measure the RHCP and LHCP components at the two frequency bands.

Best way to measure those antennas is to use an open test range and known antennas to evaluate the antenna pattern in both polarizations. For your bands you can design opposite antennas like helicals. Make a pair for each LHCP and RHCP, first test each pair for a maximum gain and with the opposite antenna the polarization suppression.
Then use the calibrated pair to test your antennas.
A test dipole can be used to determine ellipticity (if important) by slowly rotating the dipole and following the received power.